Mathematical Modeling in Finance with Stochastic Processes

2.1. SINGLE PERIOD BINOMIAL MODELS 77

portfolio will have the same value as the derivative if

φSD+ψBexp(rT) = f(SD)

φSU+ψBexp(rT) = f(SU)

The solution is

φ=

f(SU)−f(SD)

SU−SD

and

ψ=

f(SD)

Bexp(rT)

−

(f(SU)−f(SD))SD

(SU−SD)Bexp(rT)

Note that the solution requiresSU 6 =SD, but we have already assumed this
natural requirement. Without this requirement there would be no risk in the
stock, and we would not be asking the question in the first place! The value
(or price) of the portfolio, and therefore the derivative should then be

V = φS+ψB

= S

f(SU)−f(SD)

SU−SD

+B[

f(SD)

Bexp(rT)

−

(f(SU)−f(SD))SD

(SU−SD)Bexp(rT)

].

=

f(SU)−f(SD)

U−D

+

1

exp(rT)

f(SD)U−f(SU)D

(U−D)

.

We can make one final simplification that will be useful in the next section.
Define

π=

exp(rT)−D

U−D

so then

1 −π=

U−exp(rT)

U−D

so that we write the value of the derivative as

exp(−rT)[πf(SU) + (1−π)f(SD)].

(Hereπ isnot used as the mathematical constant giving the ratio of the
circumference of a circle to its diameter. Instead the Greek letter for p
suggests a similarity to the probabilityp.)
Now consider some other trader offering to sell this derivative with payoff
functionf for a priceP less thanV. Anyone could buy the derivative in
arbitrary quantity, and short the (φ,ψ) stock-bond portfolio in exactly the